Document

# Linear Kernels for Edge Deletion Problems to Immersion-Closed Graph Classes

## File

LIPIcs.ICALP.2017.57.pdf
• Filesize: 0.69 MB
• 15 pages

## Cite As

Archontia C. Giannopoulou, Michal Pilipczuk, Jean-Florent Raymond, Dimitrios M. Thilikos, and Marcin Wrochna. Linear Kernels for Edge Deletion Problems to Immersion-Closed Graph Classes. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 57:1-57:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ICALP.2017.57

## Abstract

Suppose F is a finite family of graphs. We consider the following meta-problem, called F-Immersion Deletion: given a graph G and an integer k, decide whether the deletion of at most k edges of G can result in a graph that does not contain any graph from F as an immersion. This problem is a close relative of the F-Minor Deletion problem studied by Fomin et al. [FOCS 2012], where one deletes vertices in order to remove all minor models of graphs from F. We prove that whenever all graphs from F are connected and at least one graph of F is planar and subcubic, then the F-Immersion Deletion problem admits: - a constant-factor approximation algorithm running in time O(m^3 n^3 log m) - a linear kernel that can be computed in time O(m^4 n^3 log m) and - a O(2^{O(k)} + m^4 n^3 log m)-time fixed-parameter algorithm, where n,m count the vertices and edges of the input graph. Our findings mirror those of Fomin et al. [FOCS 2012], who obtained similar results for F-Minor Deletion, under the assumption that at least one graph from F is planar. An important difference is that we are able to obtain a linear kernel for F-Immersion Deletion, while the exponent of the kernel of Fomin et al. depends heavily on the family F. In fact, this dependence is unavoidable under plausible complexity assumptions, as proven by Giannopoulou et al. [ICALP 2015]. This reveals that the kernelization complexity of F-Immersion Deletion is quite different than that of F-Minor Deletion.
##### Keywords
• Kernelization
• Approximation
• Immersion
• Protrusion
• Tree-cut width

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Rémy Belmonte, Archontia Giannopoulou, Daniel Lokshtanov, and Dimitrios M. Thilikos. The Structure of W₄-Immersion-Free Graphs. ArXiv e-prints 1602.02002, February 2016.
2. Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 25(6):1305-1317, 1996.
3. Hans L. Bodlaender, Fedor V. Fomin, Daniel Lokshtanov, Eelko Penninkx, Saket Saurabh, and Dimitrios M. Thilikos. (Meta) Kernelization. J. ACM, 63(5):44:1-44:69, November 2016. URL: http://dx.doi.org/10.1145/2973749.
4. Heather D. Booth, Rajeev Govindan, Michael A. Langston, and Siddharthan Ramachandramurthi. Fast algorithms for K_4 immersion testing. J. Algorithms, 30(2):344-378, 1999.
5. Dimitris Chatzidimitriou, Jean-Florent Raymond, Ignasi Sau, and Dimitrios M. Thilikos. An O(log OPT)-approximation for covering/packing minor models of θ_r. Algorithmica, 2017. To appear.
6. Rajesh Chitnis, Marek Cygan, Mohammad Taghi Hajiaghayi, Marcin Pilipczuk, and Michał Pilipczuk. Designing FPT algorithms for cut problems using randomized contractions. SIAM J. Comput., 45(4):1171-1229, 2016.
7. Bruno Courcelle. The Monadic Second-Order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput., 85(1):12-75, 1990.
8. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21275-3.
9. Matt Devos, Zdeněk Dvořák, Jacob Fox, Jessica McDonald, Bojan Mohar, and Diego Scheide. A minimum degree condition forcing complete graph immersion. Combinatorica, 34(3):279-298, 2014.
10. Zdeněk Dvořák and Paul Wollan. A structure theorem for strong immersions. J. Graph Theory, 83(2):152-163, 2016. URL: http://dx.doi.org/10.1002/jgt.21990.
11. Zdeněk Dvořák and Liana Yepremyan. Complete graph immersions and minimum degree. ArXiv e-prints 1512.00513, December 2015.
12. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer-Verlag, Berlin, 2006.
13. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Planar ℱ-Deletion: Approximation and optimal FPT algorithms. ArXiv e-prints 1204.4230, October 2012.
14. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Planar ℱ-deletion: Approximation, kernelization and optimal FPT algorithms. In Proceedings of FOCS 2012, pages 470-479. IEEE Computer Society, 2012.
15. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. Bidimensionality and kernels. In Proceedings of SODA 2010, pages 503-510. SIAM, 2010.
16. Robert Ganian, Eun Jung Kim, and Stefan Szeider. Algorithmic applications of tree-cut width. In Giuseppe F. Italiano, Giovanni Pighizzini, and Donald Sannella, editors, Proceedings of MFCS 2015, volume 9235 of Lecture Notes in Computer Science, pages 348-360. Springer, 2015.
17. Archontia C. Giannopoulou, Bart M. P. Jansen, Daniel Lokshtanov, and Saket Saurabh. Uniform kernelization complexity of hitting forbidden minors. ACM Trans. Algorithms, 13(3):35:1-35:35, March 2017. URL: http://dx.doi.org/10.1145/3029051.
18. Archontia C. Giannopoulou, O-joung Kwon, Jean-Florent Raymond, and Dimitrios M. Thilikos. Packing and covering immersion models of planar subcubic graphs. In Proceedings of WG 2016, pages 74-84. Springer, 2016. Preprint: ArXiv e-prints 1602.04042.
19. Archontia C. Giannopoulou, Michał Pilipczuk, Dimitrios M. Thilikos, Jean-Florent Raymond, and Marcin Wrochna. Linear kernels for edge deletion problems to immersion-closed graph classes. ArXiv e-prints 1609.07780, September 2016. URL: https://arxiv.org/abs/1609.07780.
20. Archontia C. Giannopoulou, Iosif Salem, and Dimitris Zoros. Effective computation of immersion obstructions for unions of graph classes. J. Comput. Syst. Sci., 80(1):207-216, 2014.
21. Rajeev Govindan and Siddharthan Ramachandramurthi. A weak immersion relation on graphs and its applications. Disc. Math., 230(1–3):189-206, 2001.
22. Martin Grohe, Ken-ichi Kawarabayashi, Dániel Marx, and Paul Wollan. Finding topological subgraphs is fixed-parameter tractable. In Proceedings of STOC 2011, pages 479-488. ACM, 2011.
23. Ananth V. Iyer, H. Donald Ratliff, and Gopalakrishnan Vijayan. On an edge ranking problem of trees and graphs. Discrete Appl. Math., 30(1):43-52, 1991.
24. Eun Jung Kim, Alexander Langer, Christophe Paul, Felix Reidl, Peter Rossmanith, Ignasi Sau, and Somnath Sikdar. Linear kernels and single-exponential algorithms via protrusion decompositions. In Proceedings of ICALP 2013, volume 7965 of Lecture Notes in Computer Science, pages 613-624. Springer, 2013.
25. Eun Jung Kim, Sang-il Oum, Christophe Paul, Ignasi Sau, and Dimitrios M. Thilikos. An FPT 2-approximation for tree-cut decomposition. Algorithmica, pages 1-20, 2016. URL: http://dx.doi.org/10.1007/s00453-016-0245-5.
26. Tak Wah Lam and Fung Ling Yue. Edge ranking of graphs is hard. Discrete Appl. Math., 85(1):71-86, 1998.
27. Dániel Marx. Parameterized graph separation problems. Theor. Comput. Sci., 351(3):394-406, 2006. URL: http://dx.doi.org/10.1016/j.tcs.2005.10.007.
28. Dániel Marx and Paul Wollan. Immersions in highly edge connected graphs. SIAM J. Discrete Math., 28(1):503-520, 2014.
29. Rolf Niedermeier. Invitation to fixed-parameter algorithms, volume 31 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2006.
30. N. Robertson and P.D. Seymour. Graph minors. XIII. The disjoint paths problem. J. Comb. Theory, Ser. B, 63(1):65-110, 1995.
31. Neil Robertson and Paul D. Seymour. Graph minors. V. Excluding a planar graph. J. Comb. Theory, Ser. B, 41(1):92-114, 1986.
32. Neil Robertson and Paul D. Seymour. Graph minors. XX. Wagner’s conjecture. J. Comb. Theory, Ser. B, 92(2):325-357, 2004.
33. Neil Robertson and Paul D. Seymour. Graph minors. XXIII. Nash-Williams' immersion conjecture. J. Comb. Theory, Ser. B, 100(2):181-205, 2010.
34. Paul D. Seymour and Robin Thomas. Call routing and the ratcatcher. Combinatorica, 14(2):217-241, 1994.
35. Dimitrios M. Thilikos, Maria J. Serna, and Hans L. Bodlaender. Cutwidth I: A linear time fixed parameter algorithm. J. Algorithms, 56(1):1-24, 2005.
36. Paul Wollan. The structure of graphs not admitting a fixed immersion. J. Comb. Theory, Ser. B, 110:47-66, 2015.
X

Feedback for Dagstuhl Publishing